Another proof of Wilmes' conjecture

We present a new proof of the monomial case of Wilmes' conjecture, which gives a formula for the coarsely-graded Betti numbers of the G-parking function ideal in terms of maximal parking functions of contractions of  G. Our proof is via poset topology and relies on a theorem of Gasharov, Peeva, and Welker (1999) that connects the Betti numbers of a monomial ideal to the topology of its lcm-lattice.